Salam, brother - You ask important questions, so I'll do my best to provide most accurate picture of things.
Is Islam peaceful? People confuse pacifism - meaning no war under any circumstances - to be equal to peace but Islam is not pacifism neither any ideology. Islam allows just war for legitimate causes but no open-ended aggression.
Violence is unprovoked or unnecessary provocation without any legitimate cause - Nazi killing 6 million Jews for no reason or Europeans subjugating world under ''civilizing the barbarians'' and Islam never allows for this.
In war, no women, old people, children or monks in monasteries or general civilian population is never allowed to target them - Prophet forbade this long before liberalism was invented and even captives were never tortured long before Geneva convention was invented.
In Islam to kill one innocent soul is equal to killing all of mankind - see verse 5:32.
As for cartoons, mockery and vile attitude towards Islam - we're not allowed to use force or impose vigilante justice on innocent people but be vocal and start legitimate campaign to end Islam-hating and mass propaganda to make us ''enemy of state''.
We want a level of peace to practice our own religion to degree in our own lives without state mandated ''rules'' on how we should practice our religion - ironic secularism provides less freedom to Muslims -